Either way, I get the result does not equal zero, so they are linearly dependent. What if I come across an absolute value where f'(x) = 1 would make it dependent and f'(x) = -1 would make it independent? Then what?

The following representation of may be of help to you:
Squaring x gives a positive number, even if x was negative. Then the square root acts on the positive result to give back a positive result. So whether or not x is negative, returns the positive magnitude of .

Using the chain rule we differentiate the function:

Thus the product rule gives us that:
I used the fact that to get to the second to last step.

So let's compute the Wronskian:
We can see that if and only if

AN ASIDE:
In the alternative definition of I gave above, it should be noted that I am saying that is the composition of the squaring function with the square root function . Reversing the order in which you compose the two functions (square rooting before squaring) would require restricting the domain of to , which by virtue of its smaller domain is not equal to the absolute value function. If you were to consider on the whole real line you would have to have the range of by the union of the axis of the complex plane (all pure imaginary and purely real numbers), and the result for negative x is not what we want either:
Let , then , where